If it worked like Morde it obviously wouldn't be all that difficult. Even making code for a second entity wouldn't be too terribly difficult (it could still be pretty problematic) because they would just have to add to the code that lets morde or annie move their ult around. Code aside there are still multiple reasons why having a champion that relies on 2 separate parts can cause problems.
Abilities:
There are two (well two and a half) ways this can work.
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Each part has it's own abilities that can be activated independently of the other part. ------ The closest thing to this that is already in the game are champions with 2 forms (Nidalee, Jayce, Elise, etc.). These champions have two different sets of abilities that do not share cooldowns, but they only have access to one of those ability sets at a time putting them at the same number of abilities as other champions (essentially anyway). ------ If a champion were to have 2 entities with 3 or 4 abilities a piece at the same they are essentially a 6th champion. One champion has access to 6-8 abilities at one given time compared to the 3-4 abilities other champions have.
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Activating an ability uses the abilities of both entities. So using your Q activates the Q ability of entity 1 and entity 2 at the same time. If you are controlling two entities independently of the other it would be very difficult to line up the abilites for team fights, and you would waste resources fighting with just one part (would they share a mana/energy/ferocity/etc pool or each have their own), and even if you are able to successfully position both entities you would then have 6-8 abilities compared to other champions 3-4 making it 6 against 5.
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(HALF) You make it so the second entity doesn't have abilities, but then you are essentially playing morde or annie with a second part that can move further away, and what's the fun in that?
Movement:
There are three ways I can think that this would work.
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Have a champion like kindred. The second entity is always a fixed distance from the main body. But this would make it impossible to make the awesome plays that I think you have in mind for a 2 part champion.
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Have control over the champion like morde or annie. If the second body gets too far from the main body it gets teleported back, but once again with this set up (unless you drastically increased the range the body could move there would be no way for you to make the plays you have in mind.
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Make it so the bodies can move completely independent of the other. This would allow you to do what I think you want to do and gank a lane or attack an opponent from two different sides, but by doing so you are adding a 6th champion into the mix) Not to mention keeping track of and moving both parts during and outside of a team fight would take so much effort that by the time you got both parts into the correct position the fight would be over or you would have missed the opportunity to gank.
Resources:
It is pretty obvious that the entities would share gold and items, so Mana (1.) and Health (2.).
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The only reason they would need separate mana pools is if they had two completely separate and independent ability sets. Otherwise they could share a mana pool.
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Would they share a health pool (or have a linked health pool) or have seperate health pools entirely? If they shared or had linked, then when one part dies both do. However, if they had seperate health pools then do they both have to die for the other to respawn or is the second part summoned by an ability?
Really there isn't much to explain for this part.
TLDR;
As far as code is concerned while it may be problematic and full of bugs it shouldn't be too difficult to write code that allows you to do this. The problem comes from making the champion playable. How are you supposed to fluently control two completely different entities with one keyboard and one mouse (morde and annies "familiars" are very basic for a reason)? How do you make it so the team playing this champion is playing only one champion and not essentially playing 2 making the game 6 vs. 5? And so on. While this is a very possible idea making a champion with two separate parts viable (balanced and playable) is really difficult and most likely wouldn't end up being worth the effort.